Unterschiede
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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
introduction_to_digital_systems:calc_decimal_example [2021/09/15 02:41] tfischer |
introduction_to_digital_systems:calc_decimal_example [2021/09/15 03:45] (aktuell) tfischer |
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Zeile 2: | Zeile 2: | ||
----> | ----> | ||
- | $I.\quad$ | + | Calculation example for decimal value |
<---- | <---- | ||
----> | ----> | ||
- | Idea: The number | + | Idea: The numeral |
<---- | <---- | ||
----> | ----> | ||
+ | so lets start | ||
+ | <---- | ||
+ | |||
+ | ----> | ||
+ | 1. Put space between the digits \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}:} & \color{black}{} | + | \color{black}{\text{numeral}:} & \color{black}{} |
\color{white}{\text{index}: | \color{white}{\text{index}: | ||
- | \color{white}{\text{place | + | \color{white}{\text{place |
\color{white}{} | \color{white}{} | ||
- | \color{white}{\text{numerals}:} & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} |
\color{white}{\text{result}: | \color{white}{\text{result}: | ||
\end{smallmatrix} | \end{smallmatrix} | ||
Zeile 25: | Zeile 31: | ||
----> | ----> | ||
+ | 2. Write down the index for each position. \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}:} & \color{black}{} | + | \color{black}{\text{numeral}:} & \color{black}{} |
\color{blue }{\text{index}: | \color{blue }{\text{index}: | ||
- | \color{white}{\text{place | + | \color{white}{\text{place |
\color{white}{} | \color{white}{} | ||
- | \color{white}{\text{numerals}:} & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} |
\color{white}{\text{result}: | \color{white}{\text{result}: | ||
\end{smallmatrix} | \end{smallmatrix} | ||
Zeile 40: | Zeile 48: | ||
----> | ----> | ||
+ | 3. calculate the place factor \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}:} & \color{black}{} | + | \color{black}{\text{numeral}:} & \color{black}{} |
\color{black}{\text{index}: | \color{black}{\text{index}: | ||
- | \color{blue }{\text{place | + | \color{blue }{\text{place |
\color{white}{} | \color{white}{} | ||
- | \color{white}{\text{numerals}:} & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} |
\color{white}{\text{result}: | \color{white}{\text{result}: | ||
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
<---- | <---- | ||
+ | |||
----> | ----> | ||
+ | 3. calculate the place factor \\ (= ..., thousands, hundreds, tens, ones, tenths, ...) | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{white}{\text{value}:} & \color{white}{} & \color{white}{2} & \color{white}{6} & \color{white}{5} & \color{white}{8.} & \color{white}{4} & \color{white}{7} \\ | + | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ |
- | \color{white}{\text{index}: | + | \color{black}{\text{index}: |
- | \color{white}{\text{place | + | \color{blue }{\text{place |
- | \color{white}{} & \color{white}{} & \color{white}{1000} | + | \color{blue }{} & \color{blue }{} & \color{blue }{1000} |
- | \color{white}{\text{numerals}:} & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} |
\color{white}{\text{result}: | \color{white}{\text{result}: | ||
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
<---- | <---- | ||
+ | |||
----> | ----> | ||
+ | 4. write down each digit of the numeral | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{blue }{\text{value}:} & \color{blue }{} & \color{blue }{2} & \color{blue }{6} & \color{blue }{5} & \color{blue }{8.} & \color{blue }{4} & \color{blue }{7} \\ | + | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ |
- | \color{blue }{\text{index}: | + | \color{black}{\text{index}: |
- | \color{blue }{\text{place | + | \color{black}{\text{place |
- | \color{blue }{} & \color{blue }{} & \color{blue }{1000} | + | \color{black}{} & \color{black}{} & \color{black}{1000} |
- | \color{blue }{\text{numerals}:} & \color{blue }{z_i} | + | \color{blue }{\text{digits |
- | \color{blue }{\text{calc.}:} & \color{blue }{z_i \cdot B^i} & \color{blue }{2000} | + | \color{white}{\text{place value}:} & \color{white}{z_i \cdot B^i} & \color{white}{2000} |
- | \color{blue }{\text{result}: | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 83: | Zeile 99: | ||
----> | ----> | ||
+ | 5. calculate the place value \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}:} & \color{black}{} | + | \color{black}{\text{numeral}:} & \color{black}{} |
\color{black}{\text{index}: | \color{black}{\text{index}: | ||
- | \color{black}{\text{place | + | \color{black}{\text{place |
\color{black}{} | \color{black}{} | ||
- | \color{black}{\text{numerals}:} & \color{black}{z_i} | + | \color{black}{\text{digits |
- | \color{black}{\text{calc.}:} & \color{black}{z_i \cdot B^i} & \color{black}{2000} | + | \color{blue }{\text{place value}:} & \color{blue }{z_i \cdot B^i} & \color{blue }{2000} |
- | \color{black}{\text{result}: | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
<---- | <---- | ||
- | |||
- | |||
----> | ----> | ||
- | | value || 2 | 6 | 5 | 8 , | 4 | 7 | | + | 6.Add all place values |
- | | index | $i$ | 3 | 2 | 1 | 0 | -1 | -2 | | + | |
- | | $\quad\quad$ \\ $\quad\quad$| $\quad\quad$| $\quad\quad$ | + | |
- | | $\quad\quad$| $\quad\quad$ | + | |
- | | $\quad\quad$| $\quad\quad$ | + | |
- | | $\quad\quad$ \\ $\quad\quad$| $\quad\quad$ | + | |
- | <---- | + | |
- | ----> | + | \begin{align*} |
- | | value || 2 | 6 | + | \begin{smallmatrix} |
- | | index | $i$ | + | \color{black}{\text{numeral}: |
- | | place value | $B^i$ | $\small{10^3}$ \\ $\small{1000}$ | $\small{10^2}$ \\ $\small{100}$ | $\small{10^1}$ \\ $\small{10}$ | $\small{10^0}$ \\ $\small{1}$ | $\small{10^-1}$ \\ $\small{0.10}$ | $\small{10^-2}$ \\ $\small{0.01}$ | | + | \color{black}{\text{index}:} & \color{black}{i} & \color{black}{3} & \color{black}{2} & \color{black}{1} & \color{black}{0 } & \color{black}{-1} & \color{black}{-2} \\ |
- | | digit | + | \color{black}{\text{place factor}: |
- | | calc. | + | \color{black}{} & |
- | | Result | + | \color{black}{\text{digits |
+ | \color{black}{\text{place value}: | ||
+ | \color{blue }{\text{result}: | ||
+ | \end{smallmatrix} | ||
+ | \end{align*} | ||
<---- | <---- | ||
- | ----> | ||
- | |aus (2+3)|$\color{blue}{I_p} = \color{blue}{I_m} = 0$ |$I_p$ und $I_m$ sind damit definiert| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |aus (6)|$\color{blue}{I_o} = I_1 $ |$I_o$ ist damit bekannt, wenn $I_1$ bekannt ist| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |aus (7) und (3)|$I_1 - I_2 -\color{blue}{0} = 0 $ |$\quad$| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |$\quad$|$I_1 = I_2 = I_o$ |$\quad$| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |$\quad$|$\color{blue}{I_1} = \color{blue}{I_2} = \color{blue}{I_o} $ |mit (8) und (9): $I_\boxed{}=\frac{U_\boxed{}}{R_\boxed{}}$ und (5)| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | | $\quad$ |$\frac{U_1}{R_1}= \frac{U_2}{R_2} = \frac{U_A}{R_1 + R_2}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | | (10)|$U_2= U_A\cdot\frac{R_2}{R_1+R_2}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | $II.\quad$ Betrachtung der Spannungsverstärkung | ||
- | <---- | ||
- | |||
- | ---->> | ||
- | |aus (0) |$\color{blue}{A_V}=\frac{U_A}{U_E}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
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- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
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- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
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- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << |